Перейти до основного вмісту
One-dimensional diffusion processes in bounded regions with boundary conditions and a Feller-Wentzel type conjugation condition
Kopytko Bohdan Ivanovych 1 , Shevchuk Roman Volodymyrovych 2
1 Department of Higher Mathematics, Ivan Franko National University of Lviv, Lviv, 79005, Ukraine
2 Department of Higher Mathematics, Lviv polytechnic national university, Lviv, 79007, Ukraine
Keywords: boundary conditions, Feller-Wentzel type conjugation condition
Abstract
Using an analytic method we obtain an integral representation of the two-parameter operator semigroup associated with the inhomogeneous Feller process on a closed interval whose behavior at the points of boundary as well as at some interior point is determined by the boundary conditions and the conjugation condition of Feller-Wentzell’s type, respectively.
References

[1] Feller W. The parabolic differential equations and associated semi-groups of transformations // Ann. of Math. Soc. – 1952. – 55 . – P. 468-519.

[2] Ventzel A.D. Semigroups of operators corresponding to the generalized differential operator of the second order // DAN USSR. - 1956. -111, N2. - P. 269-272.

[3] Ventzel A.D. About boundary conditions for multidimensional diffusion processes // Teoriyavebidn. i its application. - 1959. -4, N2. - P. 172-185.

[4] Skorokhod A.V. Stochastic equations for diffusion processes with boundaries // Theory of probability and its applications. - 1961. - 6, N3. - P. 287-298.

[5] Watanabe S., Ikeda N. Stochastic differential equations and diffusion processes. - Moscow: Nauka, 1986. - 456 p.

[6] Kononchuk P.P., Kopytko B.I. On a semigroup of Feller operators describing the process of diffusion on a semi-line with a nonlocal boundary condition // Mathematical Bulletin of the National Technical School - 2007 - 4 - P. 129-138.

[7] Kopytko B.I., Shevchuk R.V. Model of the process of inhomogeneous diffusion on a semi-rectangle with a general Feller-Wentzel boundary condition // Carpathian Mathematical Publications. - 2011. - 3, N2. - P. 88-99.

[8] Pilipenko A.Yu. About the Skorokhod mapping for the equations with reflection with the possibility of jump-like exit from the boundary // Ukr. mat.zhurn. - 2011. -63, N9. - P. 1241-1256.

[9] Kopytko B.I. About gluing of two diffusion processes on a straight line // In the book: Probabilistic methods of infinite-dimensional analysis. - Kiev: Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, 1980. - P. 84-101.

[10] Langer H., Schenk W. Knotting of one-dimentional Feller processes // Math. Nachr. – 1983. – 113 . – P. 151-161.

[11] Portenko M.I. Diffusion processes in media with membranes: Institute of Mathematics of NAS of Ukraine, 1995. - 199 p.

[12] Kononchuk P.P., Kopytko B.I. Semigroups of operators describing the Feller process on a straight line, glued from two diffusion processes // Probability Theory and Mathematical Statistics - 2011. - 84. - P. 86-96.

[13] Kopytko B.I., Shevchuk R.V. On pasting together two inhomogeneous diffusion processes on a line with the general Feller-Wentzell conjugation condition // Theory of Stochastic Processes – 2011. – 17 (33) , N2. – P. 55-70.

[14] Shevchuk R.V. Pasting of two one-dimensional diffusion processes // Annales Mathematicae et Informaticae – 2012. – 39 . – P. 225-239.

[15] Friedman A. Equations with partial derivatives of parabolic type. - Moscow: Mir, 1968. - 427 p.

[16] Ladyzhenskaya O.A., Solonnikov V.A., Uraltseva N.N. Linear and quasilinear equations of parabolic type. - Moscow: Nauka, 1967. - 736 p.

[17] Dynkin E.B. Markov processes.  Moscow: Fizmatgiz, 1963.  859 p.

Cite
ACS Style
Kopytko , B.I.; Shevchuk, R.V. One-dimensional diffusion processes in bounded regions with boundary conditions and a Feller-Wentzel type conjugation condition. Bukovinian Mathematical Journal. 2016, 1
AMA Style
Kopytko BI, Shevchuk RV. One-dimensional diffusion processes in bounded regions with boundary conditions and a Feller-Wentzel type conjugation condition. Bukovinian Mathematical Journal. 2016; 1(1-2).
Chicago/Turabian Style
Bohdan Ivanovych Kopytko , Roman Volodymyrovych Shevchuk. 2016. "One-dimensional diffusion processes in bounded regions with boundary conditions and a Feller-Wentzel type conjugation condition". Bukovinian Mathematical Journal. 1 no. 1-2.
Export
We use own, third-party cookies, and localStorage files to analyze web traffic and page activities. Privacy Policy Settings